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October  2011, 29(4): 1553-1571. doi: 10.3934/dcds.2011.29.1553

## Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals

 1 Fakultät für Mathematik, TU Chemnitz, D - 09107 Chemnitz, Germany, Germany 2 Mathematisches Institut, Friedrich-Schiller Universität, Ernst-Abbe-Platz 2, D - 07743 Jena, Germany

Received  January 2010 Revised  September 2010 Published  December 2010

We study measures on the real line and present various versions of what it means for such a measure to take only finitely many values. We then study perturbations of the Laplacian by such measures. Using Kotani-Remling theory, we show that the resulting operators have empty absolutely continuous spectrum if the measures are not periodic. When combined with Gordon type arguments this allows us to prove purely singular continuous spectrum for some continuum models of quasicrystals.
Citation: Steffen Klassert, Daniel Lenz, Peter Stollmann. Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1553-1571. doi: 10.3934/dcds.2011.29.1553
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##### References:
 [1] Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013 [2] Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365 [3] Adrian Tudorascu. On absolutely continuous curves of probabilities on the line. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5105-5124. doi: 10.3934/dcds.2019207 [4] Dirk Frettlöh, Christoph Richard. Dynamical properties of almost repetitive Delone sets. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 531-556. doi: 10.3934/dcds.2014.34.531 [5] Jeong-Yup Lee, Boris Solomyak. On substitution tilings and Delone sets without finite local complexity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3149-3177. doi: 10.3934/dcds.2019130 [6] Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101 [7] Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33 [8] Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35 [9] Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451 [10] Bassam Fayad, A. Windsor. A dichotomy between discrete and continuous spectrum for a class of special flows over rotations. Journal of Modern Dynamics, 2007, 1 (1) : 107-122. doi: 10.3934/jmd.2007.1.107 [11] Ronald A. Knight. Compact minimal sets in continuous recurrent flows. Conference Publications, 1998, 1998 (Special) : 397-407. doi: 10.3934/proc.1998.1998.397 [12] Bassam Fayad. Discrete and continuous spectra on laminations over Aubry-Mather sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 823-834. doi: 10.3934/dcds.2008.21.823 [13] Víctor Jiménez López, Gabriel Soler López. A topological characterization of ω-limit sets for continuous flows on the projective plane. Conference Publications, 2001, 2001 (Special) : 254-258. doi: 10.3934/proc.2001.2001.254 [14] Darren C. Ong. Orthogonal polynomials on the unit circle with quasiperiodic Verblunsky coefficients have generic purely singular continuous spectrum. Conference Publications, 2013, 2013 (special) : 605-609. doi: 10.3934/proc.2013.2013.605 [15] C. T. Cremins, G. Infante. A semilinear $A$-spectrum. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 235-242. doi: 10.3934/dcdss.2008.1.235 [16] Leonid Berlyand, Giuseppe Cardone, Yuliya Gorb, Gregory Panasenko. Asymptotic analysis of an array of closely spaced absolutely conductive inclusions. Networks & Heterogeneous Media, 2006, 1 (3) : 353-377. doi: 10.3934/nhm.2006.1.353 [17] Dmitry Dolgopyat, Dmitry Jakobson. On small gaps in the length spectrum. Journal of Modern Dynamics, 2016, 10: 339-352. doi: 10.3934/jmd.2016.10.339 [18] Natalija Sergejeva. On the unusual Fucik spectrum. Conference Publications, 2007, 2007 (Special) : 920-926. doi: 10.3934/proc.2007.2007.920 [19] Umesh V. Dubey, Vivek M. Mallick. Spectrum of some triangulated categories. Electronic Research Announcements, 2011, 18: 50-53. doi: 10.3934/era.2011.18.50 [20] Tim Alderson, Alessandro Neri. Maximum weight spectrum codes. Advances in Mathematics of Communications, 2019, 13 (1) : 101-119. doi: 10.3934/amc.2019006

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